An equation says two things are equal. Do the same to both sides and they stay equal. Solving means undoing what was done to x, one move at a time, until x stands alone.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Before any rule: which of these numbers makes 3x + 4 = 19 true?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
The order of moves
Clear any brackets by distributing. Combine like terms on each side. Gather the x terms on one side and the numbers on the other. Then undo: add or subtract first, multiply or divide last, the reverse of the order they were done in.
Letters on both sides
Move the smaller x term, so the coefficient stays positive. 5x + 3 = 2x + 12: subtract 2x from both sides and it is 3x + 3 = 12. Subtract 3: 3x = 9. Divide by 3: x = 3.
Fractions first
x/3 + 2 = 6 is easier once the fraction is gone. Multiply every term by 3: x + 6 = 18, so x = 12. Multiplying through by the common denominator clears every fraction in one move, and the rest is what you already do.
Solving for a letter
A formula is an equation with several letters. Decide which letter you want alone, and treat every other letter as a number for the duration. A = lw solved for w: divide both sides by l, and w = A/l. In d = rt, solving for t divides both sides by r: t = d/r.
The same moves, letters instead of numbers
Solve P = 2l + 2w for w. Subtract 2l from both sides: P − 2l = 2w. Divide both sides by 2: w = (P − 2l)/2. Nothing collapses to a single number, and that is the only difference from an equation full of numbers.
Check in the original
Substitute the answer into the equation you were given, never into a line you wrote yourself. A mistake made half way through survives a check in your own line. With x = 3 in 5x + 3 = 2x + 12: 15 + 3 = 18 on the left, 6 + 12 = 18 on the right. It holds.
Step 2: Try It Yourself
Tap and try it out.
Solve 3(x − 2) + 4 = 2x + 9
- 13(x − 2) + 4 = 2x + 9the equation as given
Step 0 of 5
3x + 2 = 11
3x + 2 balances 11. Take the same off both pans and they stay level, because the same was done to each side.
Step 3: In Real Life
In a weather app
A weather app turns Celsius into Fahrenheit with F = 1.8C + 32. Rearranged for C, the same equation turns Fahrenheit back. Engineers rearrange formulas like this every day, for speed, cost, dose or fuel, because the quantity they want is rarely the one the formula was written for.
Step 4: Watch an Example
One step at a time.
Watch Priya Solve 2(3x − 1) + 5 = 4x + 13
A bracket, loose numbers on the left, and x on both sides. Every move done to both sides goes in the margin.
- Step 1
The 2 reaches both terms in its bracket, 2 × 3x = 6x and 2 × 1 = 2, so the line becomes 6x − 2 + 5 = 4x + 13. Nothing was done to both sides, so the margin stays empty.
Watch Sam Solve x/4 + 5 = 2x − 2
A fraction, and x on both sides. The last three moves are yours: answer each one before it is written.
- Step 1
Every term multiplied by 4 clears the fraction: x + 20 = 8x − 8. In the margin: × 4. The 5 and the −2 are multiplied too; a fraction cleared from one term alone makes a different equation.
Two Roads: 4(2x − 3) = 20
Leo distributes first. Ana divides first. Both roads are legal, and they must meet.
- Step 1
Leo opens the bracket first. The 4 reaches both terms: 8x − 12 = 20. No margin, because nothing was done to both sides yet.
Step 5: Your Turn
Practice makes it stick.
The Tutor
Problem 1 of 2
A tutor charges a $15 booking fee for each lesson, plus $30 an hour. Maya booked two lessons of the same length and the bill came to $150. How many hours long was each lesson?
hours
The Interest
Problem 2 of 2
Simple interest is I = Prt. Ravi's $400 earned $24 interest at 2% a year, so r = 0.02. How many years was the money saved?
years
Solve and Check
Write every move in the margin. Check each answer in the original before you type it.
1 of 5
Solve 5x − 7 = 18.
2 of 5
Solve 6x + 5 = 2x + 29.
3 of 5
Solve 2(x + 5) − 3 = 3x − 2.
4 of 5
Solve x/3 + 1 = 5 − x.
5 of 5
A club sells adult tickets at $8 and child tickets at $5, so its takings are T = 8a + 5c. Solve the formula for c. Type it as c = …
Whose Step?
Each page has one wrong line. The reasons are hidden; the margins are not. Follow each move from the line above.
1 of 6
Noor solved 4x − 6 = 2x + 10. Which line has the slip?
Solve 4x − 6 = 2x + 10
- 14x − 6 = 2x + 10
- 22x − 6 = 10− 2x
- 32x = 4+ 6
- 4x = 2÷ 2
2 of 6
The same page, 4x − 6 = 2x + 10, with line 3 done right. What is x?
3 of 6
Leo solved 3(x + 5) = 2x + 21. Which line has the slip?
Solve 3(x + 5) = 2x + 21
- 13(x + 5) = 2x + 21
- 23x + 5 = 2x + 21
- 3x + 5 = 21− 2x
- 4x = 16− 5
4 of 6
Line 2 of that page, done right. The right side is untouched — what does the left side become?
5 of 6
Dev solved 3x + 6 = 27 by dividing first. Which line has the slip?
Solve 3x + 6 = 27
- 13x + 6 = 27
- 2x + 6 = 9÷ 3
- 3x = 3− 6
6 of 6
The same equation, 3x + 6 = 27, done right. What is x?
Pick the Move
No solving yet. Decide the first move, then say why.
1 of 2
For 6(x − 2) = 42, which first move is the shorter road?
2 of 2
For x/2 + x/3 = 10, which first move clears both fractions at once?
Same Solution?
Without solving. A legal move done to both sides keeps the solution; anything else changes it.
1 of 2
Which of these share their solution with 4x − 10 = 2x + 6? Select every one that does.
2 of 2
The club formula again: T = 8a + 5c. Which of these say the same thing? Select every one that does.
Step 6: Quick Check
Show what you know.
Question 1 of 2
Solve 3(x + 4) = x + 20.
Question 2 of 2
Where should you check your answer?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
For which value of k does 5(x − 2) + k = 3x + 4 have x = 0 as its solution?
Stretch 2 of 2
Find k so that 5x − k = 2x + 7 has the same solution as 3(x − 1) = 9.
What You Learned
- Clear brackets, combine like terms, gather, then undo in reverse order.
- Whatever is done to one side is done to the other. Write it in the margin.
- Rearranging a formula is the same process with letters in place of numbers.
- Check in the original equation, never in a line you wrote.
For the grown-up
The misconception this lesson targets: that a move done to one side, or to some of the terms on a side, keeps the equation true. Every slip in Whose Step? is that one idea, and the margin is the cure: a move written there was done to every term on both sides.
Why rearranging is worth doing — A formula rearranged once can be used many times. Physics, finance and engineering all state relationships in one form and need them in several. Rearranging is also how you turn a formula into a function of the variable you actually control.
Check a rearrangement with numbers — Pick numbers satisfying the original equation, then check your rearranged version gives the same relationship. If d = 60 and r = 30 give t = 2 in both forms, the rearrangement is probably right. This catches sign and division errors quickly.